A Sheafy Adjunction
26 Jul 2020
Given a topological space and a continuous map , there is an adjunction between the categories of sheaves on and sheaves on . The adjunction is somewhat mysterious to me as I begin to write this, so the hope is that by writing about it, I’ll begin to understand.
The functors in question are the pushforward and inverse image functors, which arise naturally in algebraic geometry (so I am led to believe). What makes the adjunction difficult to understand is that the definitions of the functors in question don’t seem to play nicely with each other.
The proof (with a little setting the stage) appears below. The method of proof, like most category-theoretic proofs, is a story of careful bookkeeping with functors and natural transformations, along with simple calculations to check. The idea essentially seems to be that the unit “shreds a section into pieces,” while the counit recognizes that these “shredded up” sections assemble into sections of .
Remember, given categories and , a pair of functors and form an adjoint pair when there exists an isomorphism
which is natural in and . We say is left adjoint to , or equivalently that is right adjoint to . In particular, setting , we have
which gives us a natural transformation . Similarly, we have . The natural transformation is called the unit of the adjunction, and is the counit. In fact, the existence of these natural transformations could be taken to be the definition of an adjunction, provided the natural transformations satisfy the identities
which says that for all and , the compositions
are the identity.
My goal in this blog post is to show that the functors I am interested in satisfy this latter definition of an adjunction.
Given a pair of spaces and , and a continuous map , we have functors and from the category of sheaves on to the category of sheaves on and vice versa. The former is simple to describe: if is a sheaf on , the pushforward sheaf on is the assignment
for all open subsets of . If is a morphism of sheaves on , the morphism is defined by the rule
The latter is slightly more complicated. Recall that if is a sheaf on , there is a space equipped with a local homeomorphism . The sheaf is the sheaf of local sections of ; is called the espace étale for the sheaf . The set of points of above is the stalk .
As usual, a section of the projection is an element of . In our case, continuity of is the following condition
Given that and admit maps to , we can form their pullback,
which as a set is . The topology on is induced from the product topology on . The inverse image sheaf is the sheaf of sections of . The universal property of the pullback ensures that this really defines a functor from the category of sheaves on to the category of sheaves on .
Condition for continuity of a section —that is, a tuple in —says that for all there exists an open neighborhood of and an open neighborhood in such that . Furthermore there exists a section such that for all .
If is a morphism of sheaves on , the resulting morphism is defined by
One needs to demonstrate that the right-hand side satisfies condition ; the check is simple, so I’ll leave it to the reader.
The claim is that is left adjoint to . Therefore we should expect natural transformations and .
The Unit of the Adjunction
So let be a sheaf on , and let be an open set in . Since , we can think of as a section , i.e. a tuple satisfying the compatibility condition above.
Note that in particular if is open and , the continuity condition for a section is satisfied by the section , and thus defines an element of .
Let us thus define a map as
(Let’s parse the notation: a natural transformation yields a map of sheaves for each sheaf . This is the definition of that map on the open subset .)
To prove the claim that is a natural transformation, we need to show that given a map of sheaves on , the following diagram commutes
Given , commutativity of the square is the claim that the following equality holds
This is in fact true, since taking the stalk at is functorial.
The Counit of the Adjunction
Now suppose that is a sheaf on . Let denote the espace étale over for the sheaf .
A section is a tuple taking values in satisfying the continuity condition , which in our particular case asserts the existence, for each , of an open neighborhood of and an open neighborhood in satisfying . Furthermore there exists such that the image of in under the natural map is equal to .
Since the sets cover by assumption, sheafiness of implies the existence of a unique such that is equal to the image of under the natural map for all . Let us thus define a map sending to .
We claim that defines a natural transformation from to . To prove the claim, we need to show that given a map of sheaves on , the following diagram commutes
Given , the upper right path of the square sends us first to some such that is the image of in for all , and then sends to . On the other hand, the lower left path of the square sends first to in , and then to some such that is the image of in for all .
If is an open set containing such that represents for , we have . Furthermore we know that is such that the image of in is for all . Therefore we have that the image of in is equal to , as desired.
The inverse image–pushforward adjunction
To conclude, we just need to show that two double compositions of natural transformations are the identity. The first, translated into our particular case, says that if is a sheaf on , we have that
is equal to the identity. The difficulty here seems to be largely notational. Let be an open set. Suppose is a section in . We have
First observe that the continuity condition tells us that there is, for each , an open neighborhood of and an open set containing . On this open set we have such that .
We saw above that is the section . The map recognizes that the sections satisfy the compatibility conditions to glue up to form a section in , demonstrating that the double composition is indeed the identity. So far so good.
The second double composition we need to show is equal to the identity is defined for a sheaf on as follows
If is an open set and is a section, the first map sends to the section .
The second map recognizes that the above section and agree on each stalk, and so sends our section back to .
This completes the proof that inverse image and pushforward form an adjoint pair of functors.